Catenarian Property of Mixed Polynomial and Power Series Rings Over a Pullback

Abstract

Let T be an integral domain with a maximal ideal M, ϕ: T → K: = T/M the natural surjection, and R the pullback ϕ−1(D), where D is a proper subring of K. We give necessary and sufficient conditions for the mixed extensions R[x 1]]…[x n ]] to be catenarian, where each [x i ]] is fixed as either [x i ] or [[x i ]]. We also give a complete answer to the question of determining the field extensions k ⊂ K such that the contraction map Spec(K[x 1]]…[x n ]]) → Spec(k[x 1]]…[x n ]]) is a homeomorphism. As an application, we characterize the globalized pseudo-valuation domains R such that R[x 1]]…[x n ]] is catenarian.